Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,4)(7,11)(8,12)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(5,13)(8,12)(9,15)(14,16)(18,19)(23,25)(24,26)(30,31)(34,40)(36,44)(39,46)(41,47);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385544);